These results generalize previously known results of the same type due to Brown-Guentner (2005), Baudier (2007), Baudier-Lancien (2008), and the author (2006, 2009).

One of the main steps in the proof is: each locally finite subset of an ultraproduct admits a bilipschitz embedding into . We explain how this result can be used to prove analogues of the main results for other classes of embeddings.